There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.

Draw up the probability distribution table for $N,$ the number of selected letters that are addressed to Mr Nut.

There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.

Describe one significant feature of a vertical line graph or bar chart that could be used to represent the probability distribution for $N.$

$\theta$ is a discrete random variable and $\theta \in \left(3, 4, 5, 6\right).$ Given that $P\left(\theta =q\right)=c{q}^2$ find the value of the constant $c.$

$\theta$ is a discrete random variable and $\theta \in \left(3, 4, 5, 6\right).$ Hence, find $P\left(\theta >4\right).$

Four books are randomly selected from a box containing $10$ novels, $10$ reference books and $5$ dictionaries. The random variable $N$ represents the number of novels selected.

Find the value of $P\left(N=2\right),$ correct to $3$ significant figures.

Four books are randomly selected from a box containing $10$ novels, $10$ reference books and $5$ dictionaries. The random variable $N$ represents the number of novels selected.

Without further calculation, state which of $N=0$ or $N=4$ is more likely. Explain the reasons for your answer.

In a game, a fair $4$-sided spinner with edges labelled $0,1,2$ and 3 is spun. If a player spins $1,2$ or $3,$ then that is their score. If a player scores $0,$ then they spin a fair triangular spinner with edges labelled $0,1$ and $2,$ and the number they spin is their score. Let the variable $X$ represent a player's score.

Show that $P\left(X=0\right)=\frac{1}{12}.$

In a game, a fair $4$-sided spinner with edges labelled $0,1,2$ and 3 is spun. If a player spins $1,2$ or $3,$ then that is their score. If a player scores $0,$ then they spin a fair triangular spinner with edges labelled $0,1$ and $2,$ and the number they spin is their score. Let the variable $X$ represent a player's score.

Draw up the probability distribution table for $X,$ and find the probability that $X$ is a prime number.